The graph-isomorphism conjecture for rational quadratic-form graphs

Let $q_1,q_2\boxtimes$ be quadratic forms on $\operatorname{\mathbb{Q}}^n$, and let $G_0(\operatorname{\mathbb{Q}}^n,q)$ denote the connected component containing $\operatorname{\textbf{0}}$ in $G(\operatorname{\mathbb{Q}}^n,q)$. Write $q_1\stackrel{\operatorname{\mathbb{Q}}}{\sim}q_2$ when the forms are rationally equivalent. Graph-isomorphism conjecture. If $G_0(\operatorname{\mathbb{Q}}^n,q_1)\simeq G_0(\operatorname{\mathbb{Q}}^n,q_2)$, then $q_1\stackrel{\operatorname{\mathbb{Q}}}{\sim}q_2$. Moreover, every isomorphism between these graphs is a linear transformation $T$ such that

$$q_1(\operatorname{\textbf{x}})=q_2(T\operatorname{\textbf{x}}).$$

This conjecture generalises the rational Beckman--Quarles theorem: when $q_1=q_2=I_n$, it asserts that every automorphism of $G_0(\operatorname{\mathbb{Q}}^n,I_n)$ is an isometry.